Operations with Polynomials

Identify the main elements of the following polynomial:

$P(x) = -7x^4 + 2x^3 - x + 12$

• a) What is its degree?

• b) What is the leading coefficient?

• c) What is the constant term?

• d) How many terms does it have?

Evaluate the numerical value of the polynomial $A(x) = 3x^3 - 5x^2 + 2x - 7$ for the following cases:

• a) For $x = 0$

• b) For $x = 1$

Find the value of the coefficient $k$ in the polynomial $P(x) = 2x^3 - kx^2 + 5x - 4$, knowing that evaluating its value for $x = 2$ gives a result of $10$.

Look at the following diagram of a rectangular skate park track. Write the simplified polynomial that represents the perimetre of the track in terms of $x$.

In a strategy video game, your inventory is divided into two chests. The contents of the first chest are given by the polynomial $P(x) = 5x^2 + 3x - 4$ and that of the second by $Q(x) = 2x^2 - 5x + 8$, where $x$ represents the rarity level of the items. Calculate the total polynomial representing the sum of the two chests: $P(x) + Q(x)$.

To organize an esports event, a company uses the expression $G(x) = 10x^2 - 4x + 100$ to calculate its expenses and $I(x) = 15x^2 + 2x$ to calculate its income (where $x$ is the number of sponsors). Find the polynomial that represents the net profit, knowing that profit is the difference between income and expenses: $I(x) - G(x)$.

In a video game shop, the inventory of three different categories is expressed by the following polynomials, where $x$ represents the number of new releases this month:

• Action: $A(x) = 3x^2 + 5x - 8$

• Adventure: $B(x) = -x^2 + 2x + 10$

• Strategy: $C(x) = 2x^2 - 4x + 1$

Calculate the following expressions, simplifying the result:

1. a) The total number of video games in the shop: $A(x) + B(x) + C(x)$
2. b) The difference between Action and Adventure games: $A(x) - B(x)$
3. c) How many Strategy games would be left if half of a batch of size $4x^2 - 2x$ is sold? Calculate: $C(x) - (2x^2 - x)$

In a laboratory, the growth of a bacterial population $A$ follows the polynomial $A(t) = 2t^2 + 10t + 50$, where $t$ is the time in hours. Another population $B$ grows according to the polynomial $B(t) = t^2 - 5t$.

If scientists need to know the behavior of a mixed culture defined by the operation $C(t) = 3 \times A(t) - 2 \times B(t)$, what is the simplified polynomial that represents this population $C$?

Determine the unknown polynomial $Q(x)$ in the following equality:

$(-3x^3 + 5x^2 - 7) + Q(x) = x^4 - 2x^2 + 5x - 1$

Next, calculate the numerical value of $Q(x)$ for $x = -2$.

Perform the following operation by multiplying the monomial by the polynomial (applying the distributive property):

$3x^2 \times (2x^3 - 4x + 5)$

A young designer is creating a logo for his clothing brand. The central logo is a rectangle like the one in the figure. Find the polynomial that represents the total area of the figure.

Perform the following polynomial multiplications and collect like terms:

1. a) $3x \times (2x^2 - 5x + 4)$
2. b) $(-2x^2) \times (x^3 - 3x^2 + 7)$
3. c) $(x + 3) \times (x^2 - 2x + 5)$
4. d) $(2x - 1) \times (3x^2 + x - 2)$

Calculate the area of the shaded region in the following figure. The figure represents a rectangular metal plate from which an inner rectangle has been removed to leave room for an electronic component. Express the result as a polynomial in terms of $x$.

Additional data:

• Outer base: $5x + 3$

• Outer height: $3x$

• Inner base: $x + 2$

• Inner height: $x$

Given the following combined polynomial expression, expand and simplify to obtain a single polynomial arranged in descending order of degree:

$P(x) = 2x \times (x^2 - 3x + 1) - (x - 4) \times (2x + 5)$

A parcel company designs rectangular prism-shaped boxes. The dimensions of the box are as follows: the length is $x + 5$ cm, the width is $x - 2$ cm, and the height is double the width.

1. a) Express the volume of the box as a polynomial in terms of $x$.
2. b) Expand and simplify this polynomial.

Apply algebraic identities to expand the following expressions directly, without multiplying step by step:

1. a) $(3x + 4)^2$
2. b) $(5x - 2y)^2$
3. c) $(2x^2 + 1) \times (2x^2 - 1)$
4. d) $(\frac{1}{2}x + 6)^2$

An architect designs a square town square with a side length of $L = 2x + 5$ meters. In the center, they decide to place a square fountain, also with a side length of $l = x - 3$ meters.

1. a) Expand the expression for the area of the square using algebraic identities: $(2x + 5)^2$
2. b) Expand the expression for the area of the fountain using algebraic identities: $(x - 3)^2$
3. c) Calculate the polynomial representing the area of the square that remains free for walking (Area of square - Area of fountain).

Apply algebraic identities to expand and simplify the following expression:

$3 \times (2x + \frac{1}{2})^2 - 2 \times (3x - 1) \times (3x + 1)$

Work out the following expression, simplifying the resulting polynomial as much as possible. Pay special attention to the order of operations and the use of brackets with negative signs in front of them:

$(2x^2 - 5) \times (x + 3) - (x - 4)^2 + 3x \times (x^2 - 2x + 1)$

In the center of a square garden with a side length of $2x + 5$ meters, a rectangular swimming pool has been installed with dimensions of $x + 2$ meters in length and $x - 1$ meters in width. Find the polynomial that represents the area of the lawn (the part of the garden not occupied by the pool).

A cell phone case manufacturer has determined that the cost of producing $x$ units is given by the polynomial $C(x) = 30x^2 + 600x + 12\,000$, while the revenue obtained from selling those same units is expressed as $I(x) = -12x^2 + 3\,000x$.

1. a) Find the polynomial $B(x)$ that represents the profit (Revenue minus Cost).
2. b) Calculate the profit if $40$ cell phone cases are sold.

Factorise out the common factor in the following simple algebraic expressions:

• a) $4x^2 + 8x$

• b) $5a^3 - 10a^2 + 15a$

Factorising out the common factor is the reverse process of the distributive property. Analyse the following algebraic expressions and factorise out all possible common factors (numbers and letters):

1. a) $12x^4 - 8x^3 + 4x^2$
2. b) $15a^3b^2 - 10a^2b$
3. c) $9x^2y - 3xy^2 + 6xy$
4. d) $7x^5 - 14x^2$

Simplify the following algebraic fraction by first factorising out the common factor in the numerator and the denominator:

$\frac{12x^4 - 18x^3 + 6x^2}{6x^3 - 6x^2}$